c© 2016 Cecilia Klauber

ADVANCED MODELING AND COMPUTATIONAL METHODS FORDISTRIBUTION SYSTEM STATE ESTIMATION

BY

CECILIA KLAUBER

THESIS

Submitted in partial fulfillment of the requirementsfor the degree of Master of Science in Electrical and Computer Engineering

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2016

Urbana, Illinois

Adviser:

Assistant Professor Hao Zhu

ABSTRACT

Growing penetration of distributed energy resources and smart grid tech-

nologies interfacing with the power distribution network motivate the con-

tinued advancement of accurate and robust system monitoring tools. Tra-

ditional state estimation approaches rely on iterative methods to solve the

weighted least squares problem because of the nonlinear relationship between

the power measurements and voltage phasor state. It is known that these

methods may be prone to convergence and numeric instability issues, such

as in the presence of a measurement set of diverse quality. In this thesis, dis-

tribution system state estimation techniques are developed to address this

monitoring need and take advantage of recent interest in alternative power

flow models for distribution systems and convex and quadratic optimization

methods. Therefore, semidefinite and quadratic programming methods, en-

abled by alternative power flow models, are leveraged to provide accurate

solutions that are robust to various measurement types yet computationally

efficient.

The first proposed method employs a reformulation of the power flow mea-

surement equations that captures the quadratic relationship between power

and voltage. The state estimation problem is cast as a semidefinite program

and gains the desirable convergence and solution accuracy characteristics

therein. This method attains near-optimal performance without suffering

from the numerical issues caused by variety of measurement quality, specif-

ically the inclusion of virtual measurements at zero-injection nodes. The

second method utilizes linearized power flow equations to cast the problem

as a quadratic program with linear constraints. With minimal added compu-

tational complexity, the estimate is improved by including approximations of

the nonlinear terms ignored during the linearized model development. This

method also efficiently provides a reliable state estimate while avoiding the

ill-conditioning issues that plague the traditional iterative methods. Numer-

ii

ical tests have been successfully performed on the IEEE 13-bus and 123-bus

case studies.

iii

To my family, for their love and support.

iv

ACKNOWLEDGMENTS

I would like to thank my advisor, Hao Zhu, for her support and guidance. I

am continually inspired by her tenacity, passion, and dedication to research

excellence, and I am truly grateful for the opportunity to continue my ed-

ucation under her guidance. I am grateful to the professors, students, and

staff of the University of Illinois Power and Energy Systems Group. It has

been a joy and an honor to work alongside you all.

I would like to acknowledge the College of Engineering, the Carver Foun-

dation, and the National Science Foundation Graduate Research Fellowship

for financial support of this work.

Finally, I would like to thank my friends and family for all the encourage-

ment and support over the years.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . 11.1 Motivation and Context . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Contributions and Outline . . . . . . . . . . . . . . . . 5

CHAPTER 2 SEMIDEFINITE PROGRAMMING FOR DSSE . . . . 72.1 Matrix Representation for Multi-Phase Power Flow . . . . . . 72.2 SDP-based State Estimation . . . . . . . . . . . . . . . . . . . 82.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER 3 LINEAR POWER FLOW MODELING FOR DSSE . . 123.1 Modeling of Single- and Multi-Phase Networks . . . . . . . . . 123.2 Comparison of the Linear Models . . . . . . . . . . . . . . . . 173.3 LDF-based State Estimation . . . . . . . . . . . . . . . . . . . 203.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 24

CHAPTER 4 SIMULATION RESULTS . . . . . . . . . . . . . . . . 254.1 SDP-based State Estimation Results . . . . . . . . . . . . . . 254.2 LDF-based State Estimation Results . . . . . . . . . . . . . . 294.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 35

CHAPTER 5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . 375.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

vi

LIST OF TABLES

4.1 Euclidean norm estimation error results of the SDP SEscheme for the 13-bus system . . . . . . . . . . . . . . . . . . 26

4.2 Euclidean norm estimation error results of the SDP SEscheme for the 13-bus system with bad data . . . . . . . . . . 28

4.3 The rMSE results of the four LDF-based SE schemes forthe 13-bus system . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 The rMSE results of LDF-based SE schemes for the 123-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5 Actual and estimated transformer tap ratio and positionfor one instance of the 123-bus results . . . . . . . . . . . . . . 35

vii

LIST OF FIGURES

3.1 An example of a radial feeder . . . . . . . . . . . . . . . . . . 133.2 Voltage magnitude error of single-phase model for the sim-

plified 123-bus feeder . . . . . . . . . . . . . . . . . . . . . . . 183.3 Voltage magnitude error of multi-phase model for the IEEE

13-bus feeder . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Flow chart depicting the steps of the ∆-LDF:AC method . . . 213.5 Classical transformer model and its equivalent transformer

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1 The IEEE 13-bus test feeder . . . . . . . . . . . . . . . . . . . 264.2 SDP estimation error comparison of node voltage magni-

tude for the 13-bus system . . . . . . . . . . . . . . . . . . . . 274.3 SDP estimation error comparison of node voltage angle for

the 13-bus system . . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Measurement residual at every meter in the 13-bus system

for SDP bad data detection . . . . . . . . . . . . . . . . . . . 284.5 Absolute voltage error results of three LDF SE schemes for

the 13-bus system . . . . . . . . . . . . . . . . . . . . . . . . . 314.6 Average estimation error in (left) voltage magnitude and

(right) line power flows versus the noise level in power in-jection measurements . . . . . . . . . . . . . . . . . . . . . . . 32

4.7 Average estimation error in voltage magnitude versus num-ber of measurements (out of 20) chosen to be at a highernoise variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.8 Voltage magnitude error results for the 123-bus system, byphase and distance . . . . . . . . . . . . . . . . . . . . . . . . 34

4.9 Percent of occurrences in erroneous tap estimation versusseverity of error, with and without µPMU . . . . . . . . . . . 36

viii

CHAPTER 1

INTRODUCTION

Distribution system state estimation (DSSE) can provide a nearly real-time

view of network flow and component status by processing network-wide mea-

surement data. This situational awareness enables critical support and func-

tionality such as system security, monitoring, and control [1]. With a rapidly

growing number of distributed renewable generation sources and smart grid

technologies interfacing with distribution networks, effectively obtaining the

system status in a timely manner through the development of efficient and

reliable state estimation (SE) methods is of increasing interest. Meanwhile,

these new devices and technologies also introduce additional challenges to

the DSSE problem. These obstacles include increased stress on the grid,

potential for fraud or cyber attack, and complication of operations. For ex-

ample, the installation of distributed energy resources, usage of electric ve-

hicles as energy storage, and implementation of demand response programs

contribute to the advent of less predictable flow patterns on the network.

Where power flow was once unidirectional and predictable, comprehensive

monitoring techniques are increasingly essential for effective system opera-

tions and control. In addition to the changes in physical loads and resources,

the cyber infrastructure landscape has significantly advanced with improved

sensing, communication, and control capabilities available to distribution sys-

tem operators. Hence, there is a timely opportunity to develop advanced SE

methods that can improve visibility while enhancing efficiency and reliability

for distribution systems.

1.1 Motivation and Context

For bulk transmission systems, the SE problem has been extensively inves-

tigated with several established algorithms and successful implementations

1

[2, 3]. The SE problem becomes more challenging, however, for distribu-

tion systems due to differences in network characteristics and availability of

measurements [4]. Specifically, several features of distribution systems are

known to contribute to the ill-conditioning issues of Jacobian matrices for

conventional power flow methods and applications. These include:

• Radial or weakly meshed topology

• High R/X ratio of line segments

• Unbalanced loads and phasing

Among these three, the unbalanced feature also challenges the computa-

tional efficiency of DSSE. Power transmission systems are often designed to

be well-balanced, which allows for the symmetrical component transforma-

tion of transmission lines. Thus a single-phase representation of the network

is sufficient to capture the three-phase power flow equations. In contrast,

distribution systems are inherently unbalanced. This is because line conduc-

tors may not be fully transposed, loads are not always balanced, and the

line segments can even be single-phase or double-phase. Accordingly, the

symmetrical component transformation no longer holds and a multi-phase

representation is needed for distribution system analysis. Consequently, the

dimension of the problem increases, motiving the consideration of efficient

computational methods to accelerate DSSE solvers.

Similar to transmission SE, DSSE can be formulated using the weighted

least-squares (WLS) error criterion [5]. The WLS criterion is statistically

optimal given that the measurement error is independent and Gaussian with

known variance. Under the nonlinear ac power flow equations, the resulting

WLS-SE problem is often solved using some variation of the Gauss-Newton

method. A common candidate for solving nonlinear least-squares problems,

the Gauss-Newton algorithm is rooted in taking linear approximations in

an iterative manner. Challenged by the nonconvex objective, such iterative

procedures get stuck at local minima, encounter convergence issues, or ex-

perience sensitivity to the initial guess. These issues are often worsened in

DSSE solvers due to the aforementioned problems related to ill-conditioned

power flow Jacobian matrices.

To address these convergence issues, several reformulations based on dif-

ferent variables and coordinates were considered when SE methods were first

2

developed specifically for distribution systems. Transmission SE typically re-

lates power measurements to complex voltage in polar coordinates. In [6,7],

power measurements are first converted to equivalent line current measure-

ments while the voltage phasors are represented in rectangular coordinates.

These current-based formulations capitalize on the fact that the resulting gain

matrix is constant. Hence, the computation needed to build and factorize the

gain matrix is simplified and the algorithm time can be greatly reduced. De-

spite accelerated computation, these methods may likely converge to a local

optimum in the presence of a large number of more accurate measurements

of current magnitude. Furthermore, the computational benefits associated

with the constant gain matrix will diminish for meshed distribution systems.

In [8,9], the DSSE can be formulated using branch currents in rectangular or

polar coordinates as the system states, instead of the voltages. This trans-

formation is more powerful, decoupling the system into per-phase analysis,

which would lead to smaller subproblems for each phase. Yet this method is

still sensitive to disproportional weighting factors arising from variations in

measurement accuracy, while the initialization again plays a crucial role in

the update trajectory.

In addition to the aforementioned numerical issues, the lack of accurate

or real-time measurements also challenges DSSE with system observability

issues. Although more and more sensing and communication infrastructure

have been deployed to distribution networks, the level of measurement re-

dundancy still lacks in comparison with transmission networks. Hence, it is

imperative for the SE to include all available network and load information.

Traditional metered measurements for DSSE include real and reactive power

flows and injections, voltage magnitudes, and sometimes current magnitudes.

In addition to these, DSSE have to also incorporate the following:

• pseudomeasurements

• virtual measurements

• µPMU data

Pseudomeasurements typically refer to load forecasts obtained from histor-

ical data. They may be necessary to achieve system observability, but they

can be viewed as less accurate “measurements” compared to metered data,

3

thus the much higher noise variance level. Accordingly, the noise standard

deviation of pseudomeasurements could be dozens of times that of metered

measurements.

As for virtual measurements, they correspond to network constraints at

zero-injection nodes. Typically at switching devices or branching nodes, there

is no load or generation, and thus the complex power injection exactly equals

zero. This more commonly exists in distribution systems than transmission

ones. One approach to including such constraints into the iterative DSSE

solvers is to treat them as slightly noisy data at extremely high fidelity. Es-

sentially, a much larger weight would be given to the virtual measurements

in the WLS error objective. In the presence of both virtual measurements

and pseudomeasurements, there exists a high level of variation among the

accuracy of all input information. Accordingly, the conditioning of the gain

matrix obtained by iterative linearization would degrade. Another approach

is to incorporate zero-injection bus data using Lagrange multipliers [7]. The

system of equations resulting from the problem’s necessary conditions can

be solved iteratively by Gauss-Newton as well. The challenge is that the re-

sulting system of equations is not only larger than the original unconstrained

one, but also indefinite, requiring specialized factorization techniques to en-

sure numeric stability.

Micro-phasor measurement unit (µPMU) data is becoming more available

in distribution systems along with the development of synchrophasor technol-

ogy in transmission grid monitoring. Phasor measurement units (PMUs) are

increasingly employed to provide high-resolution and synchronized samples

of voltage and current phasors. For economic and technical concerns, a more

cost-effective counterpart, the µPMU device, has been advocated for distri-

bution system monitoring [10]. A number of linear SEs have been developed

using synchrophasor measurements, including [11–14]. However, these efforts

are limited by an inability to include any other type of data, preventing their

widespread adoption. Because of the small number of µPMU devices de-

ployed currently and in the near future, it is crucial for any practical DSSE

solver to incorporate both traditional and new data types in a hybrid fashion

[15,16].

Recent research efforts in DSSE have focused on either real system imple-

mentation challenges [17–23], or extending the problem to include objectives

such as meter placement and distributed generation control [24–28] . By and

4

large, these methods still rely on iterative update procedures by linearizing

the multi-phase relationship between power and voltage phasors. Although

existing infrastructure and algorithm developments have significantly ad-

vanced monitoring capabilities in distribution networks, DSSE solvers are

still challenged by two main issues. First, the unbalanced multi-phase power

flow model leads to higher computational complexity in the linearization step.

Second, the algorithmic core of iterative updates adopted by existing solvers

fails to address the numerical conditioning issues, especially considering the

vast differences in measurement accuracy. Therefore, it is of high interest to

develop new DSSE algorithms that address these shortcomings and achieve

the dual objectives of efficiency and robustness.

1.2 Thesis Contributions and Outline

This thesis aims to develop efficient and robust DSSE approaches that can

resolve the divergence issues of existing solvers and incorporate all relevant

data and information regarding the system and its loads. Our DSSE algo-

rithms are proposed by considering two classes of multi-phase power flow

modeling approaches that are different from the existing relations between

power and voltage phasors. Specifically, the first DSSE method builds upon

reformulating the problem as a semidefinite programming (SDP) problem to

solve for a matrix system state inspired by the quadratic relationship be-

tween power and voltage quantities. The second DSSE method utilizes a

linear power flow model, relating power and voltage magnitude quantities,

to enable more efficient computation.

In contrast to the potential convergence issues encountered using standard

power flow modeling and iterative solvers, problems cast as SDPs have de-

sirable performance characteristics and can even attain the global optimum

in polynomial-time. SDP has been of recent interest in several power system

applications, including transmission system SE [29] and the optimal power

flow problem for distribution systems [30]. In this thesis, a DSSE solver is

proposed, founded on a reformulation of the power flow model which takes

advantage of the naturally linear relationship between power and voltage

squared and a relaxation of the rank constraint. This yields a convex SDP

problem which can be efficiently solved by interior-point methods, but also

5

easily incorporate virtual measurement equality constraints. Without con-

cern for convergence to a local optimum or lack of convergence thereof, sat-

isfactory approximation of the original nonconvex SE problem is expected

and demonstrated. Due to time complexity, SDP may not be appropriate for

large-scale systems, but does provide a good initialization if there is no prior

knowledge of the system.

To eliminate the need for troublesome iterative methods, linear power flow

models can also be employed. Akin to the usage of the DC power flow model

in transmission systems, a linear, but less accurate, model enables the fast

and robust analysis of large systems. The linearized DistFlow (LDF) model

[31] was a pioneer of linearized distribution system power flow models, devel-

oped to enable system operations and planning. Motivated by the influx of

interest in distribution networks, alternatives including a fixed-point based

method [32], a rectangular-coordinate formulation [33], and multi-phase un-

balanced models [30,34,35] have been recently developed. In this thesis, the

LDF model is improved upon by introducing a per-phase line flow differ-

ence term to counter the error introduced by the minimal loss assumption

that enables the linearization. Upon this linearized model, a DSSE prob-

lem is formulated by minimizing the weighted measurement mismatch error

while adhering to the zero-injection bus equality constraints. The resultant

DSSE is a quadratic optimization problem with linear constraints, for which

efficient convex optimization solvers are available as well as a closed form

solution. The LDF-based formulation is extended to include synchrophasor

measurements from µPMUs and the modeling of tap-changing voltage regu-

lators. The proposed LDF-based SE achieves an improved state estimate due

to the inclusion of the approximate line flow difference terms and exhibits

excellent numerical performance, even for a diverse set of measurements.

This thesis is organized as follows: Chapter 2 develops a multi-phase net-

work model for use with semidefinite programming for distribution system

SE. Chapter 3 proposes improvements and extensions to the linearized Dist-

Flow power flow model and formulates an SE method that utilizes the linear-

ity of the model to maintain computational tractability. Chapter 4 provides

simulation results for the proposed SE methods on the IEEE 13-bus and

123-bus systems. Chapter 5 summarizes the thesis, highlighting the key

modeling, computational, and performance benefits of the two methods and

describing intended future work.

6

CHAPTER 2

SEMIDEFINITE PROGRAMMING FORDSSE

This chapter will introduce a matrix-based multi-phase network model, for-

mulated to be linear with respect to the outer product matrix of the voltage

phasor vector. A least squares-based state estimation problem is developed

around this model, and a semidefinite relaxation is applied, rendering the

problem a convex SDP with desirable convergence and solution characteris-

tics. This chapter also addresses recovering the voltage vector state from the

outer product matrix and a process for bad data detection.

2.1 Matrix Representation for Multi-Phase Power Flow

Consider a distribution network modeled by the graph G := (N , E), and let

N := {0, ..., N} be the set of nodes connected by line segments in the set

E := {(i, j)} ⊂ N ×N . Each bus in the system may have one, two, or three

phases, each of which corresponds to a node in the set N . Let v ∈ CN×1

be the complex node voltage vector and let i ∈ CN×1 be the node current

injection vector. The network admittance matrix, Y ∈ CN×N , linearly relates

the injected currents to node voltage per Kirchhoff’s laws; that is, i = Yv.

Using the classical methods from [36] to derive the admittance matrix

for an unbalanced three-phase system, the resulting matrix contains infor-

mation about the effects of mutual coupling and branch admittances. The

approaches outlined in [36, Ch. 6,8] generalize this modeling for any line

segment or component, such as switches, voltage regulators, and transform-

ers. The shunt admittance of a distribution line is often small enough to be

ignored without loss of accuracy, though exceptions exist.

The complex power injected at node j can be expressed as

pj + jqj = vjI∗j = vj

n∑k=1

(Yjkvk)∗ (2.1)

7

where Yjk is the (j,k)-th entry of Y. Since all elements of i are linear com-

binations of vj’s, the nodal complex power pj + jqj is quadratic (not linear)

with respect to v. Consider expressing the quadratic components linearly in

terms of the outer matrix V = vvH. Because of its special structure, matrix

V is positive semidefinite; i.e., V � 0 and rank 1. To develop the linear

relation between matrix V and the complex power injections, define vector

ej to be all zeros except for the jth entry which will be 1. We define the

admittance-related matrix Yj := ejeTj Y and build the following matrices:

Hp,j :=1

2(Yj + YHj ) (2.2a)

Hq,j :=j

2(Yj −YHj ) (2.2b)

HV,j := ejeTj . (2.2c)

Using these definitions, the linear formulation of the nodal power with

respect to V becomes

pj = tr(Hp,jV) (2.3a)

qj = tr(Hq,jV) (2.3b)

|Vj|2 = tr(HV,jV) (2.3c)

where tr denotes the matrix trace operator.

2.2 SDP-based State Estimation

The purpose of state estimation is to obtain the unknown voltage magni-

tudes and angles from a set of measurements that may include the following

quantities:

• |Vj|: the voltage magnitude at node j;

• Pj: the real power injected at node j;

• Qj: the reactive power injected at node j.

8

In practice, metered measurements are known to be corrupted by noise. The

general measurement model for the i-th measurement can be described by

zi = hi(v) + εi (2.4)

where hi(v) denotes the corresponding measurement function for each zi.

Additive measurement noise εi is concatenated into a column vector ε, where

each εi ∼ N (0, σ2i ) is Gaussian distributed and independent. The set of me-

tered power injection and voltage magnitude measurements constitute the

vector z = (z1, z2, ..., zm)T , where entries z1, ..., zm can be metered measure-

ments, pseudomeasurements estimated from historical data, or a combination

thereof. The set of nonlinear functions h(v) represents the physical relation-

ship between z, v, and ε.

The goal of SE is to find an estimate of v, denoted by v, that best matches

the measurement set z according to the relationships in (2.4). To that end,

a semidefinite relaxation (SDR) approach to DSSE is considered. Since (2.3)

provides a linear relationship between the measurement values and V, as op-

posed to the quadratic relationship in (2.1), let Hi denote the corresponding

measurement matrix for each zi. Desiring to minimize the difference between

the measured and estimated data leads to the following WLS estimator:

V = arg minV∈Cn×n

m∑i=1

1

σ2i

[zi − tr(HiV)

]2s.t. tr(HiV) = 0, i = m+ 1, ...,m+ l

V � 0, and rank(V) = 1.

(2.5)

The objective captures the minimization of the difference between the mea-

sured (i = 1, ...,m) and estimated data, while the virtual measurements

(i = m + 1, ...m + l) are accounted for in the equality constraints. The

constraints on matrix V are a result of the structure of the outer product

matrix.

Although zi and V are linearly related, the reformulated problem (2.5) is

still nonconvex. The rank constraint of V is the cause of this nonconvexity.

To address this, the rank constraint is relaxed, and (2.5) is converted to a

standard SDP problem formulation. By Schur’s complement lemma [29], the

quadratic objective of (2.5) can be represented by a linear weighted cost by

9

introducing an auxiliary vector χ ∈ Rm, as given by

{V,χ} = argminV,χ

wTχ

s.t.

[χi zi − tr(HiV)

zi − tr(HiV) 1

]� 0, i = 1, ...,m

tr(HiV) = 0, i = m+ 1, ...,m+ l

V � 0

(2.6)

where the weight vector w = [ 1σ21, ..., 1

σ2m

]T .

This reformulated SE problem (2.6) is now in the standard convex SDP

form. Hence, it can be efficiently solved using convex optimization solvers.

The performance will be verified numerically in Section 4.1. Notably, the

SDP-based SE formulation incorporates the zero-injection virtual measure-

ments using equality constraints. Due to the convexity of (2.6), the highly

accurate virtual measurements will cause minimal numerical issues, as com-

pared to using the iterative approach for solving the standard problem.

The SDP-based SE formulation can conveniently incorporate additional

measurement types, which is especially valuable in networks with minimal

measurement redundancy. Line current magnitude data is commonly avail-

able in many distribution feeders, often from protection devices. The line

current magnitude squared |Ijk|2 is also quadratic with respect to v, and

hence can be linearly related to V as in (2.3). The difficulty for includ-

ing µPMU data in this framework is that the data is linear with respect to

the state v. The implementation of such measurements is not covered in

this work, but the potential is promising as PMUs have been successfully

incorporated into SDP-based SE for transmission systems in [29].

Having recovered V from the SDP solution, it remains to recover the sys-

tem state v. To do so, there must exist a one-to-one mapping between the

state vector v and the state matrix V. To prove this, let v ∈ Cn×1. The

state matrix V was defined to be vvH, which implies that for every v there

is only one V. Recall that V is positive semidefinite, symmetric, and rank

one; by the spectrum decomposition theorem V has one unique eigenvalue

and eigenvector pair. Therefore, there exists only one v =√λ1g1 such that

vvH for every V. This shows that there is a one-to-one mapping between v

and V.

10

The SDP solution V procured from (2.6) is only approximately rank-one

because of the semidefinite relaxation and subsequent removal of the rank-

one constraint in (2.6). One way to recover the state v is by using the largest

eigenvalue. Applying eigenvalue decomposition gives

V =

ρ∑i=1

λigigHi (2.7)

where ρ is the rank of matrix V, λ1 ≥ λ2 ≥ ... ≥ λp ≥ 0 are the eigenvalues of

matrix V, and g1,g2, ...gp are the corresponding eigenvectors. Since λ1g1gH1

is the best rank-one approximation to V, the SDP-based SE is v =√λ1g1.

Lastly, it is advantageous to consider bad data detection for robust SE

problems [3]. The weighted least absolute value (WLAV) error criterion is

known to handle outliers when there is sufficient measurement redundancy.

One method by which to detect bad data is to obtain the SE from a WLAV

estimator and check the residuals. A residual with large magnitude is a likely

candidate for bad data.

2.3 Chapter Summary

In this chapter, a state estimation method is developed using a matrix-based

representation of the multi-phase power flow equations and semidefinite pro-

gramming. First, the power flow measurements are uniquely reformulated

to be linear in the outer product matrix, V. Then the SDP SE problem is

introduced and relaxed, enabling computationally efficient solvers to achieve

the global optimum. A different power flow model and solution method dis-

tinguish this formulation from existing DSSE methods.

11

CHAPTER 3

LINEAR POWER FLOW MODELING FORDSSE

This chapter provides an introduction to linearized power flow models for

distribution systems. Linear power flow models have computational and

numeric stability benefits when used in the state estimation problem. The

LinDistFlow based models for single-phase networks are first introduced, then

extended to their multi-phase counterparts. Both models are written in ma-

trix form to emphasize their linear form. The resulting power flow solutions

are compared with those provided by alternative linear power flow models of

recent interest. The methods developed in this chapter are then incorporated

into efficient and robust state estimators for distribution systems.

3.1 Modeling of Single- and Multi-Phase Networks

The methods proposed in this work are a variant of the DistFlow method

formulated in [31]. Consider a distribution network modeled by the graph

G := (N , E), and let N := {0, ..., N} be the set of buses connected by

line segments in the set E := {(i, j)} ⊂ N × N . For each line (i, j), let

zij = rij + jxij denote the complex impedance and

Sij = Pij + jQij (3.1)

the complex line flow from bus i to j. Additionally, let vj, pj, and qj be

the voltage magnitude, real and reactive power injection, respectively, per

bus j. All quantities are in per unit (p.u.). Figure 3.1 is an example of a

distribution network labeled accordingly. The power flow and voltage drop

for line (i, j) is modeled by the DistFlow equations [31]:

12

Figure 3.1: An example of a radial feeder

Pij −∑k∈N+

j

Pjk = −pj + rijP 2ij +Q2

ij

v2i(3.2a)

Qij −∑k∈N+

j

Qjk = −qj + xijP 2ij +Q2

ij

v2i(3.2b)

v2i − v2j = 2(rijPij + xijQij)− (r2ij + x2ij)P 2ij +Q2

ij

v2i(3.2c)

where N+j = {k|(j, k) ∈ E with k downstream from j}. The fractional term

(P 2ij + Q2

ij)/v2i is equivalent to the squared current magnitude for line (i, j),

and contributes to the line power loss terms in equations (3.2a)- (3.2b). As-

suming negligible line flow losses, define

µi := v2i . (3.3)

Then (3.2) can be approximated by the LinDistFlow (LDF) equations [31]

as follows:

Pij −∑k∈N+

j

Pjk = −pj (3.4a)

Qij −∑k∈N+

j

Qjk = −qj (3.4b)

µi − µj = 2(rijPij + xijQij). (3.4c)

In [31], a Taylor series expansion is performed to transform (3.4c) to a linear

function of vi. The proposed method maintains the linearity in v2i to avoid the

13

loss in accuracy from an additional assumption at the minimal computational

cost of taking the square root of the elements of the solution vector.

To illustrate the linearity of (3.4), a matrix representation is introduced.

Let M0 denote the (N + 1)×N graph incidence matrix for network G. For

each line segment (i, j) ∈ E , set M0il = 1 and M0

jl = −1, if bus i is closer

to bus 0 than bus j. Then define the vector mT0 and the N × N matrix M

such that M0 = [m0 MT ]T . Stacking all quantities into vectors, the LDF

equations are equivalent to

LDF: MP = p (3.5a)

MQ = q (3.5b)

MTµ + m0 = 2(DrP + DxQ) (3.5c)

where Dr is an N × N diagonal matrix with entries corresponding to line

resistance. Similarly Dx is an N × N diagonal matrix with entries corre-

sponding to line reactance.

The source of approximation error for the LDF method is the assumption

that line flow losses are negligible. A new term is introduced to mitigate the

approximation error of (3.5) associated with the ignored fractional loss term.

First a power flow solution must be obtained using (3.5); then an adjustment

term can be introduced for each line (i, j), denoted by

lij := (P 2ij +Q2

ij)/µi. (3.6)

For each line, estimate the line current terms {lij} and substitute them into

(3.2a)-(3.2b) as constant values to form the following model:

l-LDF: MP′ = p + Drl (3.7a)

MQ′ = q + Dxl (3.7b)

MTµ′ + m0 = 2(DrP′ + DxQ

′). (3.7c)

Linearity of the l-LDF model is maintained because l is constant, by design.

The inclusion of this loss term improves the accuracy of the power flow

solution (µ′,P′,Q′), as will be shown in the following section.

In addition to the single-phase model, it is imperative to explore a multi-

phase model as well because of the unbalanced nature of distribution sys-

14

tems. Therefore this section will extend the multi-phase LDF model in [37].

An improved approximation using constant adjustment terms similar to the

method developed in the previous section for l-LDF is now introduced.

Without loss of generality, assume every bus includes three phases {a, b, c}.Each phase at each bus is referred to as a node. To account for the mutual

coupling effects between phases, the impedance of line (i, j) is represented by

the 3 × 3 matrix Zij. Accordingly, each bus voltage phasor is denoted by a

3× 1 complex vector Vj := [V aj , V

bj , V

cj ]T . Multiphase Ohm’s law states that

Vi = Vj + ZijIij (3.8)

where Iij is the 3 × 1 complex line current vector. The power flow equa-

tions (3.5a)-(3.5b) can be readily extended to model multi-phase systems by

first collecting all the per-phase line flows and node injections in correspond-

ing vectors. The matrix M also needs to be modified to maintain the flow

conservation at every node.

To obtain the voltage drop relation for line (i, j), multiply each side of

multi-phase Ohm’s law by its Hermitian conjugate and keep the real-valued

diagonal:

diag(ViVHi ) = diag(VjV

Hj )+2Re{diag(ViI

HijZHij )}+diag(ZijIijI

HijZHij ) (3.9)

Neglecting the last term which is related to line losses, as was done with the

single-phase model, the second term can be approximated by linear combi-

nations of line flows by assuming that the bus voltages are nearly balanced;

i.e.,vaivbi≈ vbivci≈ vcivai≈ φ (3.10)

where φ = ej2π/3 and φ := [1 φ φ2]T . The squared voltage µi := diag(ViVHi ),

and (3.9) can be approximated by

µi − µj ≈ 2Re{ZijSij} (3.11)

where Sij = diag(ViIHij ) is the complex line flow and Zij := diag(φ)ZHijdiag(φ∗)

15

The matrix form of the multi-phase LDF model (m-LDF) is given by

m-LDF: MP = p (3.12a)

MQ = q (3.12b)

MTµ + m0 = 2(DrP + DxQ) (3.12c)

where Dr and Dx are modified to account for the mutual coupling effect;

i.e., they are no longer diagonal matrices. They are constructed from linear

combinations of the real and reactive parts of Zij, respectively [37].

As seen in (3.5), the main source of error is due to the approximation of

flow conservation. Recall that for the single-phase case, it suffices to use

the squared line current to model the difference between the sending and

receiving line flows. This no longer holds for multi-phase lines because of

mutual coupling and additional losses on the neutral line [36]. The difference

between the sending and receiving line flows is defined by

∆ij := Sij + Sji = diag(ZijIijIHij ) (3.13)

by expansion of terms and application of Ohm’s law. Substituting the per-

phase current in the form Iaij = (Saij/Vai )∗, the multi-phase line flow difference

∆ as a function of line flows is given by

∆ij =

Zaaij

Sa∗ij S

aij

V a∗i V a

i+ Zab

ij

Sb∗ij S

aij

V b∗i V a

i+ Zac

ij

Sc∗ij S

aij

V c∗i V a

i

Zabij

Sa∗ij S

bij

V a∗i V b

i+ Zbb

ij

Sb∗ij S

bij

V b∗i V b

i+ Zbc

ij

Sc∗ij S

bij

V c∗i V b

i

Zacij

Sa∗ij S

cij

V a∗i V c

i+ Zbc

ij

Sb∗ij S

cij

V b∗i V c

i+ Zac

ij

Sc∗ij S

cij

V c∗i V c

i

≈

Zaaij S

a∗ij S

aij Zab

ij Sb∗ij S

aijφ∗ Zac

ij Sc∗ij S

aijφ

Zabij S

a∗ij S

bijφ Zbb

ij Sb∗ij S

bij Zbc

ij Sc∗ij S

bijφ∗

Zacij S

a∗ij S

cijφ∗ Zbc

ij Sb∗ij S

cijφ Zac

ij Sc∗ij S

cij

1/|V a

i |2

1/|V bi |2

1/|V ci |2

(3.14)

where the last approximation again assumes nearly balanced voltage. Since

µi = |Vi|2, (3.14) can be expressed as

∆ij = {ZHij ◦ (SijSHij )}[µi]

−1 (3.15)

where ◦ is the element-wise product operator and [•]−1 is the element-wise

inverse. Similar to the development of the l-LDF model, ∆ij can be computed

16

after obtaining the m-LDF power flow solution to better approximate the flow

conservation, leading to the following linear model:

∆-LDF: MP′ = p′ − Re(∆) (3.16a)

MQ′ = q′ − Im(∆) (3.16b)

MTµ′ + m0 = 2(DrP′ + DxQ

′). (3.16c)

As shown in 3.2, the inclusion of the approximate difference term in the ∆-

LDF model improves the accuracy of the power flow solution (µ′,P′,Q′) and

maintains the computation benefits of a linear model.

3.2 Comparison of the Linear Models

The proposed models are compared with existing linear alternatives to justify

their application to SE. The m-LDF and l-LDF models are first compared

with the recently developed linear model using fixed-point approximation

(FPA) in [32]. Second, the power flow solution provided by one Newton-

Raphson (NR) update using a flat voltage initialization is also compared.

Note that the FPA method coincides exactly with the first NR iteration in

scenarios with no shunt admittance [33]. These linear methods are tested

on a single-phase simplification of the IEEE 123-bus feeder under overloaded

conditions, as used in [32]. The MATPOWER [38] power flow program is

used to provide the benchmark voltage magnitude profile. Figure 3.2 shows

the absolute error in voltage magnitude at each node. The two LDF-based

methods outperform the FPA and NR-based methods, including in the over-

loaded regions (buses 10-35). The further improvement of the l-LDF method

is due to the addition of the estimated loss terms.

For multi-phase comparison, the m-LDF and ∆-LDF methods are analyzed

against another multi-phase solution achieved by linearizing the power flow

manifold [34]. Linearization by the method in [34] after a nonlinear change

of coordinates of the state variables is equivalent to the m-LDF model (3.12)

under the assumption of zero shunt admittances and using the flat voltage

profile as a linearization point. These methods are tested on the IEEE 13-

bus feeder [39] case with a fixed voltage regulator tap position to match the

published case. The open-source simulator OpenDSS [40] is used to provide

17

Figure 3.2: Voltage magnitude error of single-phase model for the simplified123-bus feeder in [32]

18

Figure 3.3: Voltage magnitude error of multi-phase model for the IEEE13-bus feeder

the benchmark three-phase voltage magnitude profile. Figure 3.3 shows the

per-phase absolute voltage error at every bus. The ∆-LDF approximation

provides the best performance among the three methods, mainly due to the

added ∆ij correction terms in the line flow equations. Without this correc-

tion, the other two methods underestimate the line flows and thus do not

fully capture the line voltage difference. Unfortunately the error seen near

the feeder head can be propagated by the voltage drop equation down the rest

of the feeder. These results from the single-phase and multi-phase systems

have demonstrated the importance of including the line flow difference term

in the linear power flow equations. It was also shown that the LDF-based

methods provide competitive accuracy in a computationally reasonable way,

suitable for further applications, such as SE.

19

3.3 LDF-based State Estimation

In this section, the previously introduced class of LDF models are leveraged

to develop a fast and accurate SE for distribution systems. The resultant

LDF-based SE allows for the integration of diverse data sources such as vir-

tual measurements at zero-injection buses and highly accurate synchrophasor

measurements. It will also be extended to include estimation of transformer

tap position for voltage regulators.

3.3.1 LDF-Based State Estimation

Define vector x := (µ,P,Q) to be the system state under the LDF mod-

els. Because of the linearity of the LDF-based models, all metered mea-

surements are linearly related to the system states, i.e., z = Hx + ε, with

ε ∼ N (0,R). Incorporating the m-LDF power flow equations into the

weighted least squares problem leads to the following problem:

< µ, P, Q > = arg minx

∥∥∥R− 12

[zM −Hx

]∥∥∥22

subject to MTµ + m0 = 2(DrP + DxQ)

MZP = 0

MZQ = 0

(3.17)

where MZ selects the rows of M that correspond to zero-injection buses.

Because the error cost in (3.17) is quadratic, directly incorporating the virtual

measurements as linear equality constraints does not affect the convexity

of the problem. A variety of convex solvers are available to solve (3.17)

efficiently. Indeed it is a linearly constrained quadratic program and thus a

closed-form solution exists; see, e.g., [41, 42].

This convex SE problem can be easily extended to incorporate the differ-

ence term introduced for the ∆-LDF model. Using the preliminary estimate

(µ, P, Q) obtained by (3.17), one can calculate the difference term ∆ using

(3.15). Now the real power injection measurement modeled is updated as

zp = pM + εp = MpP + Re(∆p) + εp. This modification can be captured by

a constant vector d added to the linear measurement model for z, and thus

20

Figure 3.4: Flow chart depicting the steps of the ∆-LDF:AC method

the LDF-based SE model would become

minimizex=(µ,P,Q)

∥∥∥R− 12

[zM − (Hx + d)

]∥∥∥22

subject to MTµ + m0 = 2(DrP + DxQ)

MZP = −Re(∆Z)

MZQ = −Im(∆Z)

(3.18)

where the zero-injection constraints are also updated to reflect the constant

difference term. The formulation remains a quadratic optimization prob-

lem with linear constraints that can be solved efficiently by available convex

solvers or analytically.

As a final step, results from the ∆-LDF method are used to solve the

traditional AC power flow (ACPF). The estimated P, Q and ∆ from the ∆-

LDF method are converted to p and q and used as inputs for standard AC

power flow analysis, as seen in Fig. 3.4. This approach returns the complex

voltage phasor everywhere in the system. Hence, although the proposed

LDF-SE formulation only includes the magnitude as the unknowns and not

the phase angles, it is possible to eventually estimate both as the LDF-SE

method can recover the per-phase power flows everywhere in the system.

It will henceforth be referred to as the ∆-LDF:AC method in numerical

comparisons.

3.3.2 Incorporating Diverse Measurements

The LDF-based SE approaches are now extended to incorporate available

µPMU measurements in addition to traditional meter data.

The bus voltage phasor data simply provides voltage magnitude informa-

21

tion, while line current phasor data along with the bus voltage phasor can be

converted to complex power flow measurements. Both of these types of mea-

surements can be directly integrated into the SE formulation in (3.17). By

design, they are much more accurate than traditional meter data and should

be weighted accordingly. With the ability to monitor multiple lines con-

nected to the same node with high precision, just a few µPMUs can greatly

contribute to the model observability and estimation accuracy.

Furthermore, the present SE framework can potentially incorporate the

current magnitude measurements available from protection devices. Without

the phase angle information as in µPMU data, one needs to assume knowl-

edge of power factor in order to convert these measurements to power flow

values. Approximate power factor can be obtained from historic data and the

resulting power flow measurements may have lower accuracy than if metered.

Fortunately, the proposed SE formulation is robust to high variability of data

quality, which makes it suitable for incorporating a multitude of data sources

ranging from highly accurate virtual measurements to pseudo-measurements

derived from less reliable historic data.

3.3.3 Incorporating Transformer Tap Position

Voltage regulating autotransformers with automatic tap-changing mecha-

nisms are a common component in distribution networks. They are often

ignored in literature on DSSE under the assumption of infrequent tap chang-

ing actions. Increasing system dynamics and intermittency motivates the

consideration of this pervasive component particularly in SE applications

[10].

By introducing a virtual secondary-side bus for every regulating trans-

former as developed in [43],[44], one can obtain an equivalent transformer

model that can be incorporated with the m-LDF approximation to estimate

voltage regulator tap position. Figure 3.5 shows the original transformer

model and the equivalent one. For an ideal transformer t between buses

pt and st, a virtual bus s′t is inserted in between. Equivalently, an ideal

transformer connects pt and s′t and the bus voltage relationship is given by

Vpt = atVs′t where the discrete tap ratio at takes one of the 32 values uni-

formly distributed within a rated range [a, a]. To tackle the nonlinearity

22

in modeling the line current-bus voltage relationship, the power flow across

the ideal transformer is replaced by a pair of additional injections, namely

−Spts′t and Spts′t , at buses pt and s′t, respectively. Buses pt and s′t are con-

sidered to be physically disconnected, but related by the injection variable

Spts′t . All system vectors and matrices need to be augmented to include

the virtual buses. Accordingly, the augmented matrix M can still be de-

rived from the incidence matrix, but it is no longer invertible as the graph

becomes disconnected by using the equivalent transformer model. The sys-

tem augmentation introduces additional linear equality constraints, adopted

from the ideal transformer relations, given by µs′t = a2tµpt , ppt = −ps′t , and

qpt = −qs′t , for every transformer t. Even though the values that at can take

are discrete, high granularity of the available positions enables the treatment

of the tap ratio as a continuous variable. Accordingly, the tap ratio satis-

fies a2 ≤ a2t ≤ a2, which can be captured by linear equality constraints to

bypass the complexity of having discrete constraints. By combining all the

additional constraints associated with the equivalent transformer model, the

SE problem can now be updated to

minimizex=(µ,P,Q)

∥∥∥R− 12

[zM −Hx + d

]∥∥∥22

subject to MTµ + m0 = 2(DrP + DxQ)

MZP = −Re(∆Z)

MZQ = −Im(∆Z)

ppt = −ps′t qpt = −qs′t ∀t ∈ T

µs′t≤ a2tµpt a2tµpt ≤ µs′t

∀t ∈ T

(3.19)

for the set of voltage regulators T . In this formulation the tap position at is

not estimated as a state, but it can be recovered by taking the ratio between

the estimated µs′t and µpt . Recall that vector m0 in the m-LDF equations is

a reference for the system and similar references need to be available for each

area of the disconnected graph created by introducing the virtual bus. To

provide a reference in each island of the graph, at least one voltage magnitude

measurement is required per area, per phase. Again, this problem remains

a quadratic problem with linear constraints conveniently solved by available

convex solvers.

23

zptst

+

−

Vst

+

−

Vs′t+

−Vpt

(a)

zptst

+

−

Vst

+

−

Vs′t

Spts′t−Spts′t(b)

Figure 3.5: Classical transformer model (a), and its equivalent transformermodel (b) in [44]

3.4 Chapter Summary

In this chapter, a family of LDF-based state estimators was considered. Im-

provements were made to the classic LDF formulation by a two-step process

where an initial estimate is used to approximate the nonlinear term assumed

to be small and ignored in the development of the model. Including this

term as a constant in the subsequent estimation improves the solution yet

maintains the quadratic form with linear constraints that is efficiently solved

by convex solvers. The formulation is extended to include µPMU measure-

ments as inputs and to model transformers such that the tap position can be

estimated.

24

CHAPTER 4

SIMULATION RESULTS

In this chapter, the proposed DSSE methods are demonstrated on various

IEEE distribution feeder test cases. These cases include zero-injection buses,

which provide the opportunity to exhibit the incorporation of virtual mea-

surements as linear constraints allowed by both formulations. Furthermore,

the robustness of the methods is also tested, either through the introduc-

tion of bad data or the increase in measurement noise. The system admit-

tance matrix and the complex voltage vector used as the solution reference

were procured via the distribution system modeling software OpenDSS [40].

We use the MATLAB-based optimization package CVX [45] with the solver

Mosek [46] to solve the SDP and quadratic optimization problems.

4.1 SDP-based State Estimation Results

In this section, the SDP-based state estimation method is applied to a mod-

ified IEEE 13-bus distribution feeder system as shown by Fig. 4.1 [39]. This

13-bus network has typical characteristics found in typical distribution sys-

tems, such as a tree-like topology, regulators and switches, single-phase and

two-phase lines, high R/X ratios, and unbalanced loads.

For this analysis, the distributed load along the line between buses 632

and 671 is modeled by a spot load located a third of the way down the line

at a new bus is referred to as Bus 670. The voltage regulator between the

feeder source and Bus 632 is removed.

The following measurements are collected for the 13-bus case: real and re-

active power injection measurements at load nodes and virtual measurements

at zero-injection nodes, which will be incorporated as equality constraints.

A key benefit of SDP-based SE is that the solution is not dependent on or

sensitive to the initial guess. Furthermore, it is known that an initial guess

25

Figure 4.1: The IEEE 13-bus test feeder

near the actual solution can increase the likelihood that the iterative WLS

solution will converge to the correct values. Therefore, in addition to the re-

sults of using SDP alone, the benefit of using the SDP estimate as an initial

guess, which will be referred to as the SDP-WLS approach, is also shown.

The estimation error ‖v − v‖2 is averaged over 50 realizations for both ap-

proaches. Performance results for the SDP and SDP-WLS estimators are

shown in Table 4.1 and Figs. 4.2-4.3.

To demonstrate the robustness of the SDP-based approach, a WLAV es-

timator is developed for bad data detection. To show proof of concept for

Table 4.1: Euclidean norm estimation error results of the SDP SE schemefor the 13-bus system

SDP SDP-WLS

‖v − v‖2 0.0454 0.0331Time (sec) 9.080 9.309

26

0 5 10 15 20 25 30 350.000

0.005

0.010

0.015

0.020

0.025

Node Index

Mag

nit

ud

eE

rror

(p.u

.)

SDPWLS with SDP initial guess

Figure 4.2: SDP estimation error comparison of node voltage magnitude forthe 13-bus system

0 5 10 15 20 25 30 350.000

0.200

0.400

0.600

0.800

1.000

Node Index

An

gle

Err

or(d

egre

es)

Figure 4.3: SDP estimation error comparison of node voltage angle for the13-bus system

the SDP-based WLAV approach, measurement redundancy is enhanced by

adding voltage magnitude meters. Bad data is generated by randomly pick-

ing one measurement from either a power or voltage magnitude meter and

27

0 10 20 30 40 50 60 70 80−0.5

0

0.5

1

1.5

2

2.5

Measurement Index

Residual

Figure 4.4: Measurement residual at every meter in the 13-bus system forSDP bad data detection

Table 4.2: Euclidean norm estimation error results of the SDP SE schemefor the 13-bus system with bad data

With Bad Data With Bad Data Detected/Removed

‖v − v‖2 0.3524 0.3492

multiplying its value by two. The resulting residual between the measure-

ment value and the reconstructed value is shown in Fig. 4.4 for each meter,

which identifies the outlying meter by significantly larger mismatch. After

removing this meter, the SDP-WLS estimator is applied using the remain-

ing measurements. Table 4.2 shows that the estimation error performance

has been improved after removing the meter data identified as bad based on

the residuals. This shows that with sufficient measurement redundancy, the

SDP-based WLAV estimator is effective for identifying bad data.

In this section, it was shown that SDP-based SE methods can be a robust

and effective alternative to the WLS approach, avoiding local optima and

guaranteeing convergence. The most important benefit is that it enables easy

incorporation of zero-injection virtual measurements, without potential for

convergence or conditioning ill-effects. It also provides a reliable initial guess

for a WLS estimator. The SDP formulation for a WLAV bad data detector

provides robustness to bad data in cases with sufficient observability. The

main drawback to leveraging SDP for SE is the increase in computational

time with system size. For example, the run time increase from an average

of 2.40 s to 9.08 s as the system size increases from 4 to 13 buses. This

28

motivates the development of distributed SDP algorithms for SE as seen in

[29,47].

4.2 LDF-based State Estimation Results

In this section, the proposed LDF-based state estimation techniques are

tested on the IEEE 13-bus and 123-bus systems [39]. Simulations using

LDF-based estimators on the 13-bus and 123-bus system show that solu-

tion accuracy can be improved by the inclusion of the ∆ term, as seen in

the linear model comparison, and µPMU measurements. Also shown are

the results of post-processing the estimate using an AC power flow solver,

which provides the voltage phasor vector as opposed to the magnitude alone,

as produced by standard LDF-based models. Using the 13-bus system, the

robustness of the estimator with respect to a diverse set of measurements

is shown, by providing results for the inclusion of µPMU measurements as

well as pseudomeasurements of varying accuracy and number. The modeling

and inclusion of the voltage regulator model is incorporated for the 123-bus

system only.

4.2.1 13-Bus Case

For the following tests on the 13-bus system, the available measurements

include complex power injection at every load with Gaussian noise σ = 0.02,

virtual measurements at every zero-injection node (σ = 0), and the three-

phase reference voltage at the feeder head. Five SE schemes have been tested:

(i) the m-LDF based one in (3.17), (ii) the ∆-LDF based one in (3.18), (iii)

the ∆-LDF based scheme with the additional AC power flow step, and (iv,v)

the first two schemes with a µPMU installed at Bus 632, a three-phase bus

close to the feeder head.

The DSSE performance averaged over 500 random realizations is listed

in Table 4.3. The estimator performance metric of root mean square error

(rMSE) is given by averaging the voltage magnitude squared error∑j

(vj−vj)2

over the realizations, and taking its square root. The units of rMSE follow

from the corresponding state variables in p.u. In Table 4.3, the ∆-based

29

Table 4.3: The rMSE results of the four LDF-based SE schemes for the13-bus system

rMSE m-LDF ∆-LDF ∆-LDF:AC m-LDF w/ µPMU ∆-LDF w/ µPMUv 3.08× 10−2 8.65× 10−3 5.75× 10−3 4.52× 10−3 4.18× 10−3

P 2.59× 10−2 1.45× 10−2 1.45× 10−2 2.35× 10−2 1.30× 10−2

Q 6.06× 10−2 3.28× 10−2 3.28× 10−2 3.48× 10−2 3.02× 10−2

methods provide significant improvement to the real and reactive power flow

estimates, with or without µPMU information.

In turn, the voltage magnitude estimate is also improved. For the scheme

that also involves acquiring the complex voltage phasor using AC power flow

(∆-LDF:AC), there is an increase in voltage magnitude accuracy in addition

to procuring voltage angle information. The per-phase voltage error compar-

ison is depicted in Fig. 4.5 for three of the five schemes. Figure 4.5 supports

the claim made in Section 3.2 that the line flow difference terms in the ∆-

LDF model increase the SE accuracy. Furthermore, a strategically placed

µPMU provides the greatest improvement to the SE schemes, especially in

a scenario such as this experiment where our limited measurement set only

includes power injection meters.

Our proposed LDF-based SE methods are highly robust to a variety of mea-

surement types and less affected by high variation in measurement accuracy

than standard iterative methods. The following simulations show robustness

under two conditions: (i) increasing inaccuracy of measurements and (ii) in-

creasing number of inaccurate pseudomeasurements. The first condition to

consider is varying the measurement noise level at all load measurements and

keeping the zero-injection equality constraints as in the previous tests. At

each noise level, the empirical rMSE of the voltage magnitude is averaged

over 100 random noise realizations. Figure 4.6 plots the rMSE values for

voltage magnitude and real and reactive power flows versus the noise stan-

dard deviation of the pseudomeasurements. As is expected, the estimation

performance degrades as the pseudomeasurements become more inaccurate,

but the estimator still returns a reasonable solution. As the noise level of the

pseudomeasurements increases, the improvement of the power flow estimates

decreases. This in turn lessens the potential improvement of the voltage mag-

nitude estimates at higher noise levels; this can be seen by comparing the

30

Figure 4.5: Absolute voltage error results of three LDF SE schemes for the13-bus system

31

voltage rMSEs of the LDF and ∆-LDF schemes as the noise level increases.

Figure 4.6: Average estimation error in (left) voltage magnitude and (right)line power flows versus the noise level in power injection measurements

To further show the robustness of the proposed LDF-based SE to variations

in measurement accuracy, a mix of metered and historically-based measure-

ments is considered. In this test, all metered measurements have σ = 0.02

and all pseudomeasurements have σ = 0.2. The number of load nodes cho-

sen to be less accurate pseudomeasurements instead of metered is gradually

increased, and the resultant voltage magnitude rMSE over 100 realizations is

plotted in Fig. 4.7. The estimation performance is desirable despite the in-

creasing rMSE with an increasing number of pseudomeasurements. This per-

formance degradation seems to be steady, even as the simulation approaches

the situation where the estimate is purely based on historical data. Recall

that the ∆-LDF:AC method provides voltage magnitude results similar to

those of the ∆-LDF model, but it also has the additional output of voltage

angle.

These tests on the 13-bus system show that the LDF-based methods pro-

vide a family of estimators that easily incorporate a variety of data types

without being hampered by the numerical conditioning and convergence is-

sues present in the typical WLS formulations.

4.2.2 123-Bus Case

The 123-bus test feeder case [39] is a radial system with multiple voltage

regulators and shunt capacitors. The system contains overhead and under-

32

Figure 4.7: Average estimation error in voltage magnitude versus number ofmeasurements (out of 20) chosen to be at a higher noise variance

ground lines, unbalanced loading, and multiple switches. As in the 13-bus

tests, power injection measurements are assumed to be available at every

load and voltage measurements at a three-phase voltage meter downstream

from each voltage regulator. As long as there is a voltage reference in each

disconnected graph created by the addition of the virtual secondary bus from

the equivalent voltage regulator model, the system will solve. For an initial

example, the voltage magnitude errors for the m-LDF and ∆-LDF models

are plotted in Fig. 4.8 under noise-free conditions. The data is separated

by phase and organized by the nodal distance from the feeder head. Note

that the error is significantly higher at phase c. Though not the heaviest

loaded phase, its loads, and thereby its line flows, have higher power factor

than the other two phases, which seems to affect the accuracy of the model

approximation.

In addition to the two SE schemes, m-LDF and ∆-LDF as seen in Fig.

4.8, the first two schemes are simulated with an additional µPMU installed

at Bus 14, a three-phase bus near the feeder head that branches into three

major feeders. Table 4.4 shows the empirical rMSE for the various schemes

over 200 random realizations.

To demonstrate the capability of estimating the transformer tap positions,

the proposed methods are implemented on the system using the introduced

equivalent models. After procuring the estimate states, the transformer tap

position can be recovered by simply dividing the secondary side voltage by

the primary side voltage for each transformer, while the tap position is pro-

33

Figure 4.8: Voltage magnitude error results for the 123-bus system, byphase and distance

Table 4.4: The rMSE results of LDF-based SE schemes for the 123-bussystem

rMSE m-LDF ∆-LDF m-LDF w/ µPMU ∆-LDF w/ µPMUv 3.58× 10−2 3.27× 10−2 2.10× 10−2 2.08× 10−2

P 1.97× 100 1.69× 100 1.72× 100 1.63× 100

Q 2.55× 100 2.16× 100 2.23× 100 2.08× 100

vided by rounding the estimated ratio to the nearest discrete ratio. For one

instance of the 123-bus results, Table 4.5 shows the actual and estimated

transformer tap ratio and position by bus and phase. In this typical exam-

ple, of the nine transformer taps in this scenario, only two were estimated

to be different than their actual position. The effects of µPMU data on

estimating the tap positions are also investigated. Figure 4.9 plots the per-

centage of the time that the estimation led to a certain number of incorrect

tap positions, shown along the x-axis. It also compares the scenario of no

µPMU data with that of having µPMU data at Bus 14, as before. This

chart shows that if the estimation was incorrect, it was likely to be close

(less than three positions off). Additional analysis shows that the estima-

tion is more likely to be incorrect when the voltage references are far from

each other, as the error accumulates over distance. This chart shows that

introducing additional measurements, like those from a µPMU, can greatly

increase the ability to accurately estimate the tap position, demonstrating

34

Table 4.5: Actual and estimated transformer tap ratio and position for oneinstance of the 123-bus results

Bus Phase Tap Ratio (Position) Estimated Ratio (Position) Difference150 a 1.0375 (6) 1.0377 (6) 0.0002

b 1.0375 (6) 1.0377 (6) 0.0002c 1.0375 (6) 1.0376 (6) 0.0001

9 a 1.0000 (0) 0.9971 (0) 0.002925 a 1.0125 (2) 1.0082 (1) 0.0043

c 1.0000 (0) 0.9984 (0) 0.0015160 a 1.0625 (10) 1.0611 (10) 0.0014

b 1.0250 (4) 1.0251 (4) 0.0000c 1.0375 (6) 1.0340 (5) 0.0035

the value that even one µPMU can have in increasing the accuracy of the

proposed distribution system SE.

The simulation results in this section show how the proposed models are

effective for large-scale systems, provide flexibility with respect to measure-

ment type, and ultimately effectively monitor the system state leveraging

all available measurement information without inhibitive concerns such as

solution convergence issues.

4.3 Chapter Summary

In this chapter, two DSSE methods were tested on various IEEE distribution

feeder systems. Both exercised their specialized modeling and solution tech-

niques to provide accurate estimates from a variety of input measurement sets

in a manner that is both computationally efficient and immune to iteration-

based ill-conditioning issues. In addition, the SDP-based method extension

to bad data detection was validated as well as the LDF-based extension to

estimating transformer tap position. These methods greatly benefit the task

of monitoring the changing distribution system by enabling the integration

of a diverse available measurement set without the consequences of numeric

ill-conditioning and lack of convergence.

35

Figure 4.9: Percent of occurrences in erroneous tap estimation versusseverity of error, with and without µPMU

36

CHAPTER 5

CONCLUSION

Motivated by the increasing need for effective monitoring tools for distribu-

tion networks, this thesis explores models and methods for distribution sys-

tem power flow and state estimation. In an effort to take into consideration

the unique characteristics and available measurements in distribution sys-

tems, two new multi-phase state estimation methods are proposed. Where

many methods maintain the nonlinear power flow equations and solve the

DSSE using iterative methods, the proposed techniques employ alternative

power flow models to enable the use of semidefinite and quadratic program-

ming solutions.

The methods are tested on the IEEE 13-bus and 123-bus systems under

various conditions, including with bad data and varied measurement sets.

The SDP-based state estimation method can seamlessly integrate virtual

measurements using equality constraints and provide an estimate via convex

solvers that is highly accurate on its own, or can be used as an initial guess

for WLS. The LDF-based state estimation technique incorporates a constant

approximate line difference term to improve upon the state estimate while

maintaining the conveniently solvable form of the problem as a quadratic

program with linear constraints. This method was also shown to successfully

assimilate a variety of measurements, including metered, virtual, pseudo, and

µPMU data.

Compared to the typical models and iterative methods, which are sensitive

to factors that may cause ill-conditioning and solution divergence, the SDP-

based and LDF-based DSSE methods successfully cast the SE problem as a

semidefinite or quadratic program and thus provide a reliable, accurate, and

robust state estimate in a computationally effective manner.

37

5.1 Future Work

Although these schemes have many advantages, additional improvements can

be explored to further increase computational efficiency and robustness as the

distribution network continues to evolve. For the SDP method, distributed

algorithms will be explored to decrease the computation time, especially as

the system size increases. Future work on the LDF methods will include scal-

ing the problem to larger, more realistic systems and further analysis of how

to best incorporate the widely-available current magnitude measurements.

38

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